THE UNIVERSITY OF NEW SOUTH WALESSCHOOL OF MATHEMATICS AND STATISTICSMATH 1231MATHEMATICS 1B ALGEBRA.Section 3: - Eigenvectors and Eigenvalues.1. Motivation and DefinitionsConsider the effect of multiplying vectors inR2by the matrixA=±1210².For exampleA±11²=±31²andA±21²=±42²01230123xAx012340123xAxIn thefirst case, we see that the vector±11²gets stretched and rotated, but in the secondcase, the vector±21²just gets ‘stretched’ by a factor of 2.Now look whatAdoes to the vectors±−10²and±−11².A±−10²=±−1−1²andA±−11²=±1−1².xAxAxxOnce again we see that±−10²is moved around, while the image of the±−11²is paralleland of the same length, but in the opposite direction. We can think of this as a ‘stretch’ by1
the factor−1.The vectors that got ‘stretched’ (possibly in a negative sense) are called theeigenvectorsfor the matrixAand the ‘stretch factor’ (in the above examples they were 2 and−1 ) arecalled theeigenvaluesof the matrix. These eigenvectors and eigenvalues play an extremelycrucial role throughout many areas of mathematics and its applications in Engineering andPhysics.You might like to compare this with what happens when we differentiateeλx. The derivativeisλeλx. In other words, when we differentiateeλx, this function gets ‘stretched’ by a factorofλ. We will exploit this idea later when looking at systems of differential equations. In themeantime, let us try to gather together what we have done and see what the implications are.